Similar to many mathematical fields also the topic of mathematical programming has its origin in applied problems. But, in contrast to other branches of mathematics, we don't have to dig too deeply into the past centuries to find their roots. The historical tree of mathematical programming, starting from its conceptual roots to its present shape, is remarkably short, and to quote Isaak Newton, we can say: "We are standing on the shoulders of giants". The goal of this paper is to describe briefly the historical growth of mathematical programming from its beginnings to the seventies of the last century and to review its basic ideas for a broad audience. During this process we will demonstrate that optimization is a natural way of thinking which follows some extremal principles.
For a given graph G and a positive integer r the r-path graph, $P_r(G)$, has for vertices the set of all paths of length r in G. Two vertices are adjacent when the intersection of the corresponding paths forms a path of length r-1, and their union forms either a cycle or a path of length k+1 in G. Let $P^k_r(G)$ be the k-iteration of r-path graph operator on a connected graph G. Let H be a subgraph of $P^k_r(G)$. The k-history $P^{-k}_r(H)$ is a subgraph of G that is induced by all edges that take part in the recursive definition of H. We present some general properties of k-histories and give a complete characterization of graphs that are k-histories of vertices of 2-path graph operator.
The expression „geographical course of history”, currently seen as old-fashioned or obsolete, seems to be typical in geographical works of Nałkowski. It combines two fields that were very close to him just from the very beginning of his scientific career: geography, to which he has devoted his whole life, and history, which was for him a source of information necessary to explain the diversity of landscapes. The origins of the concept can be found in works of German geographer, educator and philosopher Ernst Christian Kapp, which were poorly known in the Russian language area. Nałkowski had an opportunity to acknowledge Kapp’s instructions during his studies in Leipzig. The scholar wrote about three political-geographical circles of culture (germ. Political-geographische Kulturkreise), which also correspond with the stages in the development of civilization. For Nałkowski „the geographical course of history” is a specific point of view on history. Its essence is the movement of global (political and economic) center of civilization in three successive stages of social evolution. The first stage is connected with the great rivers of the ancient East, the second corresponds with the classical Greece and Rome (the Mediterranean), and the third is the period of dominance of the so-called West (the Atlantic stage). The Atlantic stage was to be replaced with the Pacific stage, before the advent of the global era. From the dawn of history until today the center of civilization shifted westward and this direction is the constant element of the concept. The article includes the results of empirical, geo-strategic and macroeconomic research, which confirm the validity of Nałkowski’s predictions. His concept became an inspiration not only to historians of civilization such as Feliks Koneczny and Eugeniusz Kwiatkowski, but also indirectly influenced the well-known theories of George Modelski, Peter Taylor and Samuel Huntington.
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